Project/Area Number  10640115 
Research Category 
GrantinAid for Scientific Research (C).

Section  一般 
Research Field 
General mathematics (including Probability theory/Statistical mathematics)

Research Institution  KYOTO UNIVERSITY 
Principal Investigator 
KOKUBU Hiroshi Kyoto Univ., Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (50202057)

CoInvestigator(Kenkyūbuntansha) 
TAKAISHI Takeshi Hiroshima Faculty of Engineering Lecturer Kokusai Gakuin Univ., 工学部, 講師 (00268666)
KOMURO Motomasa Teikyo Univ. Department of Electronics Associate Professor Sci. & Tech. and Information Sciences, 電子情報科学科, 助教授 (00186818)
ISO Yusuke Kyoto Univ., Graduate School of Professor Informatics, 大学院・情報学研究科, 教授 (70203065)
NISHIDA Takaaki Kyoto Univ., Graduate School of Science, Professor, 大学院・理学研究科, 教授 (70026110)

Project Fiscal Year 
1998 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥3,300,000 (Direct Cost : ¥3,300,000)
Fiscal Year 1999 : ¥1,800,000 (Direct Cost : ¥1,800,000)
Fiscal Year 1998 : ¥1,500,000 (Direct Cost : ¥1,500,000)

Keywords  dynamical systems / bifurcation / invariant set / topological invariant / singularity / chaos / chaotic itinerancy / attractor / 力学系 / 分岐 / 不変集合 / 位相的不変量 / 特異点 / カオス / カオス的遍歴 / アトラクタ / ベクトル場 / ホモクリニック軌道 / 大域的 / 数値計算 / 計算機支援 
Research Abstract 
The results obtained in the project are as follows: 1. In relation to a topological invariant of dynamical systems called the Conley index, the transition matrix, which plays an important role in the study of bifurcations of connecting orbits, is generalized for multiparameter families. As the first step toward developing the Conley index theory for certain types of singularly perturbed vector fields called slowfast systems, a topologicalalgebraic condition for the existence of periodic and heteroclinic orbits is obtained when the slow manifold is normally hyperbolic and onedimensional. 2. It is proved that certain type of heteroclinic cycles bifurcates from a codimension three degenerate singularity of vector fields. As a result, the occurrence of chaotic attractors from the degenerate singularity follows. 3. Stability of stationary solutions (Couette flows) in fluid motion in rotating cylinders is numerically studied when the cylinders rotate in the same/opposite directions. The problem can be reduced to ODE systems and the validated numerical simulation can be applied. As a result, the critical Taylor number is determined and, by making use of the local bifurcation theory, the existence of Taylor vortex (stationary solution) and bifurcations of periodic solutions (in some cases, wavy Taylor vortex) is rigorously proved. 4. Transition phenomena among semistable states through chaos in coupled systems of chaotic oscillators is discovered and named as "chaotic itinerancy" by Kaneko, Tsuda and Ikeda. In this project, the creation mechanism of chaotic itinerancy in globally coupled maps (GCM) is studied. From the invariance of GCM with respect to permutations of oscillators, the hiererchical structure of invariant subspaces is determined and is used to prove that the chaotic itinerancy in GCM occurs through crisis in attractors lying in lower dimensional subspaces which destabilize in the complementary directions.
